Skip to main content

Some properties of meromorphically multivalent functions

Abstract

By using the method of differential subordinations, we derive certain properties of meromorphically multivalent functions.

2010 Mathematics Subject Classification: 30C45; 30C55.

1 Introduction

Let Σ(p) denotes the class of meromorphically multivalent functions f(z) of the form

f z = z - p + k = 1 a k - p z k - p p N = 1 , 2 , 3 , ,
(1.1)

which are analytic in the punctured unit disk

U * = z : z C and 0 < z < 1 = U \ 0 .

Let f(z) and g(z) be analytic in U. Then, we say that f(z) is subordinate to g(z) in U, written f(z) g(z), if there exists an analytic function w(z) in U, such that |w(z)| ≤ |z| and f(z) = g(w(z)) (z U). If g(z) is univalent in U, then the subordination f(z) g(z) is equivalent to f(0) = g(0) and f(U) g(U).

Let p(z) = 1 + p1z + ... be analytic in U. Then for -1 ≤ B < A ≤ 1, it is clear that

p z 1 + A z 1 + B z z U
(1.2)

if and only if

p z - 1 - A B 1 - B 2 < A - B 1 - B 2 - 1 < B < A 1 ; z U
(1.3)

and

Re p z > 1 - A 2 B = - 1 ; z U .
(1.4)

Recently, several authors (see, e.g., [17]) considered some interesting properties of meromorphically multivalent functions. In the present article, we aim at proving some subordination properties for the class Σ(p).

To derive our results, we need the following lemmas.

Lemma 1 (see [8]. Let h(z) be analytic and starlike univalent in U with h(0) = 0. If g(z) is analytic in U and zg'(z) h(z), then

g z g 0 + 0 z h t t d t .

Lemma 2 (see [9]. Let p(z) be analytic and nonconstant in U with p(0) = 1. If 0 < | z0 | < 1 and Re p z 0 = min z z 0 Re p z , then

z 0 p z 0 - 1 - p z 0 2 2 1 - Re p z 0 .

2 Main results

Our first result is contained in the following.

Theorem 1. Let α ( 0 , 1 2 ] and β (0,1). If f(z) Σ(p) satisfies f(z) ≠ 0 (z U*) and

z - p f z z f z f z + p < δ z U ,
(2.1)

where δ is the minimum positive root of the equation

α sin π β 2 x 2 - x + 1 - α sin π β 2 = 0 ,
(2.2)

then

arg f z z - p - α < π 2 β z U .
(2.3)

The bound β is the best possible for each α ( 0 , 1 2 ] .

Proof. Let

g x = α sin π β 2 x 2 - x + 1 - α sin π β 2 .
(2.4)

We can see that the Equation (2.2) has two positive roots. Since g(0) > 0 and g(1) < 0, we have

0 < α 1 - α δ δ < 1 .
(2.5)

Put

f z z - p = α + 1 - α p z .
(2.6)

Then from the assumption of the theorem, we see that p(z) is analytic in U with p(0) = 1 and α + (1 - α)p(z) ≠ 0 for all z U. Taking the logarithmic differentiations in both sides of (2.6), we get

z f z f z + p = 1 - α z p z α + 1 - α p z
(2.7)

and

z - p f z z f z f z + p = 1 - α z p z α + 1 - α p z 2
(2.8)

for all z U. Thus the inequality (2.1) is equivalent to

1 - α z p z α + 1 - α p z 2 δ z .
(2.9)

By using Lemma 1, (2.9) leads to

0 z 1 - α p t α + 1 - α p t 2 d t δ z

or to

1 - 1 α + 1 - a p z δ z .
(2.10)

In view of (2.5), (2.10) can be written as

p z 1 + α 1 - α δ z 1 - δ z .
(2.11)

Now by taking A= α 1 - α δ and B = -δ in (1.2) and (1.3), we have

arg f z z - p - α = arg p z < arcsin δ 1 - α + α δ 2 = π 2 β

for all z U because of g(δ) = 0. This proves (2.3).

Next, we consider the function f(z) defined by

f z = z - p 1 - δ z z U * .

It is easy to see that

z - p f z z f z f z + p = δ z < δ z U .

Since

f z z - p - α = 1 - α 1 + α 1 - α δ z 1 - δ z ,

it follows from (1.3) that

sup z U arg f z z - p - α = arcsin δ 1 - α + α δ 2 = π 2 β .

Hence, we conclude that the bound β is the best possible for each α ( 0 , 1 2 ] .

Next, we derive the following.

Theorem 2. If f(z) Σ(p) satisfies f(z) ≠ 0 (z U*) and

Re z - p f z z f z f z + p < γ z U ,
(2.12)

where

0 < γ < 1 2 log 2 ,
(2.13)

then

Re z - p f z > 1 - 2 γ log 2 z U .
(2.14)

The bound in (2.14) is sharp.

Proof. Let

p z = f z z - p .
(2.15)

Then p(z) is analytic in U with p(0) = 1 and p(z) ≠ 0 for z U. In view of (2.15) and (2.12), we have

1 - z p z γ p 2 z 1 + z 1 - z ,

i.e.,

z 1 p z 2 γ z 1 - z .

Now by using Lemma 1, we obtain

1 p z 1 - 2 γ log 1 - z .
(2.16)

Since the function 1 - 2γ log(1 - z) is convex univalent in U and

Re 1 - 2 γ log 1 - z > 1 - 2 γ log 2 z U ,

from (2.16), we get the inequality (2.14).

To show that the bound in (2.14) cannot be increased, we consider

f z = z - p 1 - 2 γ log 1 - z z U * .

It is easy to verify that the function f(z) satisfies the inequality (2.12). On the other hand, we have

Re z - p f z 1 - 2 γ log 2

as z → -1. Now the proof of the theorem is complete.

Finally, we discuss the following theorem.

Theorem 3. Let f(z) Σ(p) with f(z) ≠ 0 (z U*). If

Im z f z f z f z z - p - λ < λ λ + 2 p z U
(2.17)

for some λ(λ > 0), then

Re f z z - p > 0 z U .
(2.18)

Proof. Let us define the analytic function p(z) in U by

f z z - p = p z .

Then p(0) = 1, p(z) ≠ 0 (z U) and

z f z f z f z z - p - λ = p z - λ z p z p z - p z U .
(2.19)

Suppose that there exists a point z0(0 < | z0 | < 1) such that

Re p z > 0 z < z 0 and p z 0 = i β ,
(2.20)

where β is real and β ≠ 0. Then, applying Lemma 2, we get

z 0 p z 0 - 1 + β 2 2 .
(2.21)

Thus it follows from (2.19), (2.20), and (2.21) that

I 0 = Im z 0 f z 0 f z 0 f z 0 z 0 - p - λ = - p β + λ β z 0 p z 0 .
(2.22)

In view of λ > 0, from (2.21) and (2.22) we obtain

I 0 - λ + λ + 2 p β 2 2 β λ λ + 2 p β < 0
(2.23)

and

I 0 - λ + λ + 2 p β 2 2 β - λ λ + 2 p β > 0 .
(2.24)

But both (2.23) and (2.24) contradict the assumption (2.17). Therefore, we have Rep(z) > 0 for all z U. This shows that (2.18) holds true.

References

  1. Ali RM, Ravichandran V, Seenivasagan N: Subordination and superordination of the Liu-Srivastava linear operator on meromorphically functions. Bull Malays Math Sci Soc 2008, 31: 193–207.

    MathSciNet  Google Scholar 

  2. Aouf MK: Certain subclasses of meromprphically multivalent functions associated with generalized hypergeometric function. Comput Math Appl 2008, 55: 494–509.

    MathSciNet  Article  Google Scholar 

  3. Cho NE, Kwon OS, Srivastava HM: A class of integral operators preserving subordination and superordination for meromorphic functions. Appl Math Comput 2007, 193: 463–474.

    MathSciNet  Article  Google Scholar 

  4. Liu J-L, Srivastava HM: A linear operator and associated families of meromorphically multivalent functions J. Math Anal Appl 2001, 259: 566–581.

    MathSciNet  Article  Google Scholar 

  5. Liu J-L, Srivastava HM: Classes of meromorphically multivalent functions associated with the generalized hypergeometric function. Math Comput Model 2004, 39: 21–34.

    MathSciNet  Article  Google Scholar 

  6. Wang Z-G, Jiang Y-P, Srivastava HM: Some subclasses of meromorphically multivalent functions associated with the generalized hypergeometric function Comput. Math Appl 2009, 57: 571–586.

    MathSciNet  Google Scholar 

  7. Wang Z-G, Sun Y, Zhang Z-H: Certain classes of meromorphically multivalent functions. Comput Math Appl 2009, 58: 1408–1417.

    MathSciNet  Article  Google Scholar 

  8. Suffridge TJ: Some remarks on convex maps of the unit disk. Duke Math J 1970, 37: 775–777.

    MathSciNet  Article  Google Scholar 

  9. Miller SS, Mocanu PT: Second order differential inequalities in the complex plane. J Math Anal Appl 1978, 65: 289–305.

    MathSciNet  Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Jin-Lin Liu.

Additional information

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

All authors read and approved the final manuscript.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Xu, YH., Yang, Q. & Liu, JL. Some properties of meromorphically multivalent functions. J Inequal Appl 2012, 86 (2012). https://doi.org/10.1186/1029-242X-2012-86

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1186/1029-242X-2012-86

Keywords

  • analytic function
  • meromorphically multivalent function
  • subordination